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A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes. 12
0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, 34, 42, 38, 36, 46, 42, 42, 50, 46, 47, 53, 50, 45, 56, 54, 46, 62, 53, 48, 64, 59, 55, 68, 61, 59, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Ways of writing 2n+1 as p+q+r where p,q,r are odd primes with p <= q <= r.
The two papers of Helfgott appear to provide a proof of the Odd Goldbach Conjecture that every odd number greater than five is the sum of three primes. (The paper is still being reviewed.) - Peter Luschny, May 18 2013; N. J. A. Sloane, May 19 2013
REFERENCES
George E. Andrews, Number Theory (NY, Dover, 1994), page 111.
Ivars Peterson, The Mathematical Tourist (NY, W. H. Freeman, 1998, pages 35-37.
Paulo Ribenboim, "VI, Goldbach's famous conjecture," The New Book of Prime Number Records, 3rd ed. (NY, Springer, 1996), pages 291-299.
LINKS
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014.
EXAMPLE
a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.
MAPLE
A007963 := proc(n)
local a, i, j, k, p, q, r ;
a := 0 ;
for i from 2 do
p := ithprime(i) ;
for j from i do
q := ithprime(j) ;
for k from j do
r := ithprime(k) ;
if p+q+r = 2*n+1 then
a := a+1 ;
elif p+q+r > 2*n+1 then
break;
end if;
end do:
if p+2*q > 2*n+1 then
break;
end if;
end do:
if 3*p > 2*n+1 then
break;
end if;
end do:
return a;
end proc:
seq(A007963(n), n=0..30) ; # R. J. Mathar, Sep 06 2014
MATHEMATICA
nn = 75; ps = Prime[Range[2, nn + 1]]; c = Flatten[Table[If[i >= j >= k, i + j + k, 0], {i, ps}, {j, ps}, {k, ps}]]; Join[{0, 0, 0, 0}, Transpose[Take[Rest[Sort[Tally[c]]], nn+2]][[2]]] (* T. D. Noe, Apr 08 2014 *)
PROG
(Sage)
def A007963(n):
c = 0
for p in Partitions(n, length = 3):
b = True
for t in p:
b = is_prime(t) and t > 2
if not b: break
if b : c = c + 1
return c
[A007963(2*n+1) for n in (0..77)] # Peter Luschny, May 18 2013
(Perl) use ntheory ":all"; sub a007963 { my($n, $c)=(shift, 0); forpart { $c++ if vecall { is_prime($_) } @_; } $n, {n=>3, amin=>3}; $c; }
say "$_ ", a007963(2*$_+1) for 0..100; # Dana Jacobsen, Mar 19 2017
(PARI) a(n)=my(k=2*n+1, s, t); forprime(p=(k+2)\3, k-6, t=k-p; forprime(q=t\2, min(t-3, p), if(isprime(t-q), s++))); s \\ Charles R Greathouse IV, Mar 20 2017
CROSSREFS
Cf. A068307, A087916, A294294 (lower bound of scatterplot), A294357, A294358 (records).
Sequence in context: A248868 A320614 A030566 * A137222 A077641 A329547
KEYWORD
nonn
AUTHOR
R. Muller
EXTENSIONS
Corrected and extended by David W. Wilson
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)